1.2. TH E CHOW-RASHEVSKY'S ACCESSIBILITY THEORE M AND CC METRICS 9 condition is needed. In this respect we emphasize that in [MM04(II)] Monti and Morbidelli have proved that in the general Hormander case any C°°, non- characteristic bounded domain is NTA with respect to the CC metric, hence in particular it is (e,£). Since every non-characteristic domain is trivially of type 2, as a consequence of these considerations, the main result in [MM02] becomes a special case of Theorem 10.7. On the other hand, it must be said that the assumption that dft be non-characteristic in [MM02] allows to avoid resorting to the extension procedure by working directly on ft. Monti and Morbidelli also treat an example of characteristic domain for the special situation of the Baouendi- Grushin vector fields in the plane X\ = d/dx, X2 \x\ad/dy, a 0. For a discussion of the latter we refer the reader to Chapter 3. 1.1. Carnot-Caratheodory spaces In this chapter we collect some definitions and various basic known results which are used in the main body of the paper. 1.2. The Chow-Rashevsky's accessibility theorem and CC metrics Let X = {Xi,..., Xm} be a system of C°° vector fields in IRn, n 3, satisfying the finite rank condition (1.4). A piecewise Cl curve 7 : [0,T] E n is called sub-unitary if for every t £ (0, T) for which ^'(t) exists one has n (1.11) V(*U2 J2XJ^W^2 for every ( G l n . The reader should notice that definition (1.11) forces the condition i{t) e span{X1(7(t)),...,X1(1(t))}. We define the sub-unitary length of 7 as /s(7) = T. Given x,y £ Rn, denote by Su(x,y) the collection of all sub-unitary 7 : [0, T] U which join x to y. We will need the following fundamental accessibility theorem due Chow [Ch39] and Rashevsky [Ra38]. THEOREM 1.4. Given a connected open set U C Rn, for every x,y G U there exists 7 £ Su{x,y). As a consequence of Theorem 1.4, if we pose (1.12) du(x,y) = inf {/s(7) | 7 G Su(x,y)}, we obtain a distance on [/, called the Carnot-Caratheodory distance on U associated with the system X. When U = Mn, we write S(x,y), instead of S^n(x,y), and d(x,y), instead of d^n(x,y). It is clear that (1.13) d(x,y) du{x,y) x,y G 17, for every connected open set U C Rn. For x £ Rn, and r 0, we let B(x,r) = {y £ Rn I d(x,y) r}. We indicate with Be(x,r) = {y £ Rn | \x - y\ r} the
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